Unit III Free Electron Theory Engineering Physics

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1 . Introduction The electron theory of metals aims to explain the structure and properties of solids through their electronic structure. The electron theory is applicable to all solids i.e., both metals and non metals. It explains the electrical, thermal and magnetic properties of solids etc. The theory has been developed in three main stages. The classical free electron theory Drude and Lorentz proposed this theory in 900. According to this theory, the metals containing the free electrons obey the laws of classical mechanics. The quantum free electron theory Somerfield developed this theory in 928. According to this theory the free electrons obey quantum laws. According to this theory the free electrons are moving in a constant potential. The zone theory Bloch stated this theory in 928. According to this theory, the free electrons move in a periodic field provided by the lattice. According to this theory the free electrons are moving in a constant potential. 2. Definitions and relations Drift velocity It is defined as the average velocity acquired by the free electrons of a metal in a particular direction by the application of electric field. Relaxation time It is defined as the time taken by the free electrons to reach its equilibrium position from the disturbed position in the presence of electric field. Collision time (τ ) It is defined as the average time taken by the free electrons between two successive collisions. Current density ( j ) It is defined as the magnitude of current passing through unit area. I J = or I = A J ds Expression for Relaxation time τ ) ( r When the metal is subjected to an external electric field, the electrons move opposite to the applied field. After removal of electric field the drift velocity decays exponentially as t τ r vd = voe Where v o is the initial velocity of an electron, before application of electric field andτ r is the relaxation time. Dr. P.Sreenivasula Reddy M.Sc, (PhD) Website: Page

2 If t τ = then r v = v e d o or v d = Dr. P.Sreenivasula Reddy M.Sc, (PhD) Website: Page 2 vo e Thus the relaxation time may be stated as the time taken for the drift velocity to decay to of its original initial value. e Mean free path (λ) Free electrons in a metal are continuously moving in all directions and with various speeds. They frequently collide with one another. Therefore, they move in straight line with constant speeds between two successive collisions. The distance traveled by the electron between two successive collisions is called as free path and their mean is called the mean free path. The average distance traveled by the electron between two successive collisions is called mean free path. Or The mean free path is the average distance traveled by an electron between two successive collisions with other free electrons. λ c τ = Where c is the mean square velocity of electron. c = 3KBT m Expression for Mean collision time:- The average time taken by the electrons between two consecutive collisions of electron with the lattice points is called mean collision time. If v be the total velocity of electron i.e. thermal and drift velocity, then the mean collision time is given by λ τ = v v = v d + v th If v d << vth then v = vth λ τ = v th Expression for drift velocity When electric field is applied on an electric charge e, then it moves in opposite direction to the field with a velocity v d. This velocity is known as drift velocity. The Lorentz force acting on the electron is F L = ee The resistance force acting on the electron is Fr = When the system is in steady state mv d r = τ ( ee) v d eeτ r = m F = F r L mv d τ r

3 Mobility of electrons The mobility of electrons is defined as the magnitude of drift velocity acquired by the electron in a unit field. µ = v d E J I neav nevd We know σ = = d = neµ E AE AE E = 2 σ n e τ eτ µ = = = ne m n e m Q I = ne Av = d 2 ne τ Q σ = m 3. The classical free electron theory of metals (Drude Lorentz theory of metals) Drude and Lorentz proposed this theory in 900. According to this theory, the metals containing the free electrons obey the laws of classical mechanics. Assumptions ( or salient features) in classical free electron theory The classical free electron theory is based on the following postulates.. The valence electrons of atoms are free to move about the whole volume of the metal, like the molecules of a perfect gas in a container. 2. The free electrons move in random direction and collide with either positive ions fixed to the lattice or the other free electrons. All the collisions are elastic in nature i.e., there is no loss of energy. 3. The momentum of free electrons obeys the laws of the classical kinetic theory of gases. 4. The electron velocities in a metal obey classical Maxwell-Boltzman distribution of velocities. 5. When the electric field is applied to the metal, the free electrons are accelerated in the direction opposite to the direction of applied electric field. 6. The mutual repulsion among the electrons is ignored, so that they move in all the directions with all possible velocities. 7. In the absence of the field, the energy associated with an electron at temperature T is given by Here 3 kt. It is related to the kinetic energy equation 2 v th represents the thermal velocity. Success of classical free electron theory Dr. P.Sreenivasula Reddy M.Sc, (PhD) Website: Page 3 3 kt = m 2 2. It verifies ohm s law 2. It explains electrical conductivity of metals. 3. It explains thermal conductivity of metals. 4. It derives Widemann Franz law. (I.e. the relation between electrical and thermal conductivity. Draw backs of classical free electron theory.. It could not explain the photoelectric effect, Compton Effect and black body radiation. 2. Electrical conductivity of semiconductors and insulators could not be explained. K 3. Widemann Franz law ( = constant) is not applicable at lower temperatures. σ T 2 v th

4 4. Ferromagnetism could not be explained by this theory. The theoretical value of paramagnetic susceptibility is greater than the experimental value. 5. According to classical free electron theory the specific heat of metals is given by 4.5R where as the experimental value is given by 3 R 6. According to classical free electron the electronic specific heat is equal to 3 R 2 while the actual value is 0.0R 4. Classical free electron theory electrical conductivity The classical free electron theory was proposed by Drude and Lorentz. According to this theory the electrons are moving freely and randomly moving in the entire volume of the metal like gas atoms in the gas container. When an electric field is applied the free electrons gets accelerated. When an electric field is applied between the two ends of a metal of area of cross section A Force acting on the electron in the electric field From Newton s second law The acceleration of electron The average velocity acquired (i.e. drift velocity) by the electrons by the application of electric field is. 2 Where = RMS velocity. The relation between current and drift velocity is 3 Substituting the value of in from equation () into equation (3), we get Conductivity Resistivity According to kinetic theory of gasses 3 Dr. P.Sreenivasula Reddy M.Sc, (PhD) Website: Page 4

5 Mobility 3 5. Sources of electrical resistance in metals According to quantum free electron theory, the free electrons always collides with the positive ions or electrons present in the metal. The scattering of conduction electrons are due to. Effect of temperature 2. Defect, e.g. impurities, imperfections, etc. Temperature effect: The positive ions are always in oscillating (or vibrating) state about their mean position; even the substance is present at 0k temperature. The vibrating amplitude of ions is always depends the temperature. The mean free path λ of the electrons is inversely proportional to the mean square of amplitude of ionic vibrations. () The energy of lattice vibrations is proportional to and increases linearly with temperature T. T (2) From equations () and (2) (3) The resistivity of the metal is inversely proportional to mean free path of electrons. (4) From equations (3) and (4) T (5) From equation (5) we observe that the resistivity of metal is linearly increases with temperature. The conductivity is defined as the reciprocal of resistance. (6) From equation (6) we observe that the conductivity of metal is inversely proportional to their temperature. The variation of resistivity of metal with temperature is shown in figure.

6 Defect dependence: The variation of resistivity of copper- nickel alloys as a function of temperature is shown in figure At a particular temperature the ideal resistivity of alloy can be written as AX X Where X is the concentration and A is a constant which depends upon the base metal and impurity. The total resistivity change is generally given by the Matthiessen s rule Where is temperature dependent resistivity due to thermal vibrations of the positive ions while is caused by the scattering of electrons by impurity atoms. is dominated at higher temperature and is dominated at low temperature. 6. Distribution laws introduction Three statistical distributions have been developed in explaining the distribution of atoms or molecules or electrons in energy levels.. Maxwell Boltzmann distribution 2. Bose Einstein distribution 3. Fermi Dirac distribution Maxwell Boltzmann distribution In Maxwell Boltzmann distribution all the particles in the system are distinguishable and no more restrictions on filling the particles in the energy levels. This is mainly applicable for atoms and molecules. According to Maxwell Boltzmann distribution, the probability of occupying an energy level is

7 Bose Einstein distribution In Bose Einstein distribution all the particles in the system are indistinguishable and no more restrictions on filling the particles in the energy levels. This is mainly applicable for bosons. (Bosons are the particles with zero or integral spin). According to Bose Einstein distribution, the probability of occupying an energy level is Fermi Dirac distribution In Fermi Dirac distribution all the particles in the system are indistinguishable and it obeys Pauli s exclusive principle (i.e., not more than two electrons can occupy the same energy level) on filling the particles in the energy levels. This is mainly applicable for fermions (Fermions are the particles with odd half integral spins 0,,,,. ). According to Fermi - Dirac distribution, the probability of occupying an energy level is Fermi energy The energy of the highest occupied level at 0K is called the Fermi energy and the energy level is referred as the Fermi level. The Fermi energy is denoted by E F Or The Fermi level is that energy level for which the probability of occupation is 2 or 50% at any temperature. Dr. P.Sreenivasula Reddy M.Sc, (PhD) Website: Page 7

8 7. Fermi Dirac distribution In Fermi Dirac distribution all the particles in the system are indistinguishable and it obeys Pauli s exclusive principle (i.e., not more than two electrons can occupy the same energy level) on filling the particles in the energy levels. This is applicable for fermions (Fermions (e.g. electrons, protons, neutrons, etc.) are the particles with odd half integral spins0,,,,. ). Fϵ. According to Fermi - Dirac distribution, the probability of electron occupying an energy level is is the energy of i th level is the energy of Fermi level, k is a Boltzmann constant T is the absolute temperature and is the Fermi function The variation of Fermi function with temperature is shown in figure ϵf Effect of temperature on Fermi Dirac distribution When the material is at a temperature T 0K, below the Fermi energy levels are totally fulfilled and above the energy levels are empty. When it receives thermal energy from surroundings, the electrons are thermally excited into higher energy levels. The occupation of electrons in energy levels obeys a Fermi Dirac statistical distribution law. According to Fermi - Dirac distribution, the probability of electron occupying an energy level is ϵ Dr. P.Sreenivasula Reddy M.Sc, (PhD) Website: Page 8

9 The variation of Fermi function with temperature is shown in figure. Fϵ ϵf ϵ From this curve we can observe the probability of finding an electron at different temperatures. Case I. At 0, 2 2 Case II. At 0, Case III. At 0, From figure () we illustrate that When 0 0

10 When 0, So the Fermi level can be defined as temperature. The energy level for which the probability of occupation is 50% at any Properties of Fermi function. It is applicable for all insulators, semiconductors and metals. 2. In semiconductor the probability of electron occupying an energy level is 3. In semiconductor the probability of hole occupying an energy level is 4. At 0, 5. At 0, 0 6. At 0, From the Fermi Dirac distribution we understood that, below the Fermi levels are totally fulfilled and above the levels are totally empty 8. Kroning and Penney model According to Kroning and Penney the electrons move in a periodic square well potential. This potential is produced by the positive ions (ionized atoms) in the lattice. The potential is zero near to the nucleus of positive ions and maximum between the adjacent nuclei. The variation of potential is shown in figure. V=V 0 II V=0 I -b 0 a The corresponding Schrödinger wave equations for I region is Ψ Ψ 0 0 Ψ Ψ 0 () (2) Dr. P.Sreenivasula Reddy M.Sc, (PhD) Website: Page 0

11 The corresponding Schrödinger wave equations for II region is Ψ Ψ 0 Ψ Ψ 0 Ψ Ψ 0 (3) (4) The general solutions of the equations () and (3) are of the form Ψ x Ae Be (5) Ψ x Ae Be (6) Solving the above equations (5) and (6) by applying boundary conditions, we get. Where potential barrier strength The plot of verses is shown in figure. From the above spectrum we observe that. The energy spectrum of the electron consists of a large number of allowed energy bands and forbidden energy bands 2. The width of the allowed energy bands increases with increasing of i.e. with increases energy. 3. As the allowed regions becomes infinity narrow and the energy spectrum becomes a line spectrum as shown in figure (2)

12 4. For 0 the energy spectrum is continuous as shown in figure (3) 9. Introduction about Band Theory and definitions The origin of an energy gap is explained by considering the formation of energy bands in solids. The concentration of atoms in a gaseous medium is very low compared to the concentrations of atoms in a solid medium. As a result, the interaction between any two atoms in a gaseous substance is very weak, since the interatomic distance is very large. In case of solid substance interatomic distance is very small, and hence there is an interaction between any two successive atoms. Due to this interaction the energy levels of all atoms overlap with each other and hence bands are formed. Band A set of closely packed energy levels is called as band Valence band A band which is occupied by the valence electrons is called as valence band. The valence band may be partially or completely filled up depending on the nature of the material. Conduction band The lowest unfilled energy band is called as conduction band. This band may be empty of partially filled. In conduction band the electrons can move freely.

13 Forbidden energy gap or Forbidden gap The energy gap between valence band and conduction band is called forbidden energy gap or forbidden gap or band gap. 0. Origin of bands in solids According to quantum free electron theory, the free electrons move in constant potential and the atoms have independent energy levels (discrete energy levels). Pauli s exclusive principle is applied in filling the energy levels. According to Kroning and Penney model the free electrons moves in a square well periodic potential and predicts (tell in advance) the existence of allowed energy band and forbidden energy gaps. The origin of an energy gap is explained by considering the formation of energy bands in solids. The concentration of atoms in a gaseous medium is very low compared to the concentrations of atoms in a solid medium. As a result, the interaction between any two atoms in a gaseous substance is very weak, since the interatomic distance is very large. In case of solid substance interatomic distance is very small, and hence there is an interaction between any two successive atoms. Due to this interaction the energy levels of all atoms overlap with each other and hence bands are formed.. Conductors, Semi conductors and Insulators The solids are classified into three types based on the forbidden gap. Those are Conductors, Semiconductors and Insulators. Conductors In case of conductors, the valence and conduction bands are overlap to each other as shown in figure.

14 In conduction band, plenty of free electrons are available for conduction. The energy gap in conductors is zero. The charge carriers in conductors are electrons. Conductors have negative temperature coefficient. All metals are the examples of conductors. Semiconductors In case of semiconductors the forbidden gap is very small as shown in figure. Germanium and Silicon are the best examples for semiconductors. In Germanium the forbidden gap is 0.7eV. In silicon the forbidden energy gap is.ev. At 0K the valence band is completely fulfilled and the conduction band is totally empty. When a small amount of energy is supplied, the electrons can easily jump from valence band to conduction band. The charge carriers in semiconductors are both electrons and holes. Insulators In case of insulators, the forbidden energy gap is very wide as shown in figure. In insulators the valence electrons are bound vary tightly to their parent atoms. In case of insulators the forbidden energy gap is always >6eV. Due tot this fact electrons cannot jump from valence band to conduction band.

15 2. Quantum free electron theory-electrical conductivity To overcome the draw backs of classical free electron theory, Sommerfeld proposed quantum free electron theory. He treated electron as a quantum particle. The electrons follow quantum laws. Under this, the velocities of the electrons are plotted in the velocity space with dots as shown in figure. The Fermi level electrons have maximum velocity and are known as Fermi velocity and is represented by the Fermi sphere. If electric field E is applied along direction, then the electrons in the sphere experiences a force along. only those electrons present near the Fermi surface can take electrical energy and occupy higher vacant energy levels. For the rest of the electrons, the energy supplied by electrical force is too small so that they are unable to occupy higher vacant energy levels. The relation between momentum and wave vector is 2 Differentiating w.r.t. t we get 2 Force applied on the electron in the applied field is given by Integrating the above the equation, we get Dr. P.Sreenivasula Reddy M.Sc, (PhD) Website: Page 5

16 On Fermi surface and The current density From equation () Substituting the value of from equation (3) into equation (4), we get 3 4 Comparing the above equations The above equation represents the electrical conductivity. Merits of quantum free electron theory. It successfully explains the electrical and thermal conductivity of metals. 2. We can explain the Thermionic phenomenon. 3. Temperature dependence of conductivity of metals can be explained by this theory. 4. It can explain the specific heat of metals. 5. It explains magnetic susceptibility of metals. Demerits of quantum free electron theory. It is unable to explain the metallic properties of exhibited by only certain crystals 2. It is unable to explain why the atomic arrays in metallic crystals should prefer certain structures only Dr. P.Sreenivasula Reddy M.Sc, (PhD) Website: Page 6

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